We say that a set \(\mathcal {A}\)  of graphs is bridge-addable if for every graph \(G \in \mathcal {A}\)  with vertices \(u\)  and \(v\)  in distinct components, the graph \(G+ uv\)  is in \(\mathcal {A}\) , where \(G+uv\)  denotes the graph obtained from \(G\)  by adding an edge between \(u\)  and \(v\) .
For example, for every surface \(S\)  the class \(\mathcal {E}^S\)  of graphs embeddable in \(S\)  is bridge-addable.
A set of graphs is decomposable when a graph is in the set if and only if each component is. We call the set addable if it is decomposable and bridge-addable (following [1]}, [2]}). It is straightforward to check that a minor-closed class is addable if and only if its excluded minors are 2-connected. Thus Corollary REF  concerns an addable minor-closed class \(\mathcal {A}\) .
